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Let $A$ be a positive matrix. What is known about the Birkhoff norm of its Perron vector?

By the Birkhoff norm of a vector $x$ I refer to the quantity $\frac{\max{x}}{\min{x}}$.

P.S. This is actually a spiced-up restatement of my older question.

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How is the Birkhoff norm of a vector defined? –  Lee Mosher Jul 3 '12 at 14:37
Thanks, I added a definition. –  Felix Goldberg Jul 3 '12 at 15:00
If I understand correctly, the Perron–Frobenius theorem asserts that the unique largest eigenvalue of $A$ has as its corresponding eigenvector a vector $x$ with entirely positive entries ($x$ is determined up to non-zero real scalar), and you are asking for the ratio of the largest to least entry of $x$? Anyway, I don't see how this is different from the older linked question. –  Theo Johnson-Freyd Aug 27 '13 at 6:58
@TheoJohnson-Freyd It's different in the way I was approaching the subject the first and the second time. The first time I was thinking in elementary terms and the second time I realized that perhaps the question could be better addressed by couching it in more general language. Whether this in retrospect merited a separate question or a major edit I am not sure, but for me it felt like a practically new one in real time. –  Felix Goldberg Aug 27 '13 at 7:08

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